Question

select the correct answer.
what is the height, x, of the equilateral triangle shown?
image of an equilateral triangle with all angles 60 degrees, base 10 in, and height labeled x
a. 5 in.
b. $5\sqrt{3}$ in.
c. 10 in.
d. $10\sqrt{3}$ in.

Explanation:

Step1: Split the equilateral triangle

An equilateral triangle with side length \( s = 10 \) in. When we draw the height \( x \), it splits the triangle into two 30 - 60 - 90 right triangles. The base of each right triangle is \( \frac{10}{2}=5 \) in, and the hypotenuse is equal to the side length of the equilateral triangle, which is 10 in.

Step2: Apply the Pythagorean theorem

For a 30 - 60 - 90 right triangle, we can also use the properties of special right triangles or the Pythagorean theorem \( a^{2}+b^{2}=c^{2} \), where \( c = 10 \) (hypotenuse), \( a = 5 \) (one leg), and \( b=x \) (the other leg, which is the height of the equilateral triangle).
Substitute the values into the formula: \( 5^{2}+x^{2}=10^{2} \)
\( 25 + x^{2}=100 \)
Subtract 25 from both sides: \( x^{2}=100 - 25=75 \)
Take the square root of both sides: \( x=\sqrt{75}=\sqrt{25\times3}=5\sqrt{3} \) in.
Or, using the property of 30 - 60 - 90 triangles, where the side opposite 60° (the height in this case) is \( \frac{\sqrt{3}}{2}\times \) hypotenuse. The hypotenuse is 10, so \( x=\frac{\sqrt{3}}{2}\times10 = 5\sqrt{3} \) in.

Answer:

B. \( 5\sqrt{3} \) in.